I agree on the unpopulated areas, I walk in them (or rubbish weather in more popular areas) simply because I'm less likely to come across someone else and those you do seem to have better control over better trained dogs and who don't have their mind blown by seeing another dog or person
I think this bit is the key though -
Obviously the dog still needs to be under control and, if there are livestock or nesting birds etc, still on a lead.
Unfortunately the under control seems to be subjective to a lot of people, you should be able to call the dog back at any given moment and direct towards or away from certain things, but of course the owner also needs to respect what certain things the dog should be kept away from - and I find that's at least some of the problem, the owner sees no problem with, or isn't engaging with the dog and noticing, it running up to other dogs or people, horses, livestock, or investigating whatever it wants and so even if it's trained, and would come back with direction from the owner, it's not given so the dog carries on.
Of course then there's those who don't have control over their dogs because they don't put the time and effort in to train it in, so a lead is the only way they can have some control and as someone who's recently rehomed a dog with zero training and a like for buggering off, it's hard work and takes a long time and consistency to achieve it, but it's also hard work to walk a dog like that on a lead, I've done it for two years and it could be very tempting just to let them off and what happens, happens and then it just becomes a pattern because it's easier for the person with the dog like that, but can affect everyone else that they come into contact with negatively.
So I can see why saying on a lead only is the only way to ensure that any dog is somewhat under control, and usually by the time you find out the other dog is not under control, either through lack of training or will from the owner, it's too late and the damage is done.
And I also agree with other posters that the dog owners who aren't already causing a problem by and large, are the ones who will adhere to any new rules around dogs in shared spaces, and the people causing the problems won't because they don't see it as a problem.